Why Use an Early Time-to-Event Outcome?
In a multi-arm multi-stage (MAMS) trial, an interim decision may be needed before the final endpoint has had enough time to occur. An earlier time-to-event outcome can provide information for deciding which experimental arms should continue. For example, progression-free survival (PFS) may be observed earlier than overall survival (OS). PFS can therefore help guide arm selection while OS remains the distinct endpoint for the trial’s final evaluation.
The distinction matters. An early endpoint can be useful for selection without being a validated surrogate for the final endpoint. A favorable PFS result does not, by itself, establish a favorable OS effect. The selection rule should be specified in advance, and participants should continue to be followed for the final endpoint according to the protocol, including after progression when the final endpoint is OS.
For PFS, the event definition commonly includes progression or death, but the protocol must specify how progression is assessed and how dates are assigned. The time origin, event definition, censoring rules, and horizon \(\tau\) should be the same across arms. A fixed-horizon event-free probability is often easy to communicate: a difference of 0.10 at 6 months means a 10-percentage-point difference in the probability of remaining event-free through 6 months.
Estimating Event-Free Probability at an Interim Look
Not every participant will have an observed event by the interim analysis. Some will be known to be event-free only until their last adequate assessment or contact; their follow-up is right-censored. Under independent censoring within treatment arm (or conditional on the variables used in the analysis), the Kaplan–Meier estimator uses the available follow-up without counting a censored participant as having an event.
The product form follows by breaking survival through \(t\) into conditional probabilities of surviving each interval. If \(q_j\) is the conditional probability of remaining event-free past event time \(t_j\), then \(S(t)=\prod_{t_j\leq t}q_j\). The observed fraction surviving among those at risk at \(t_j\), \(1-d_j/n_j\), estimates \(q_j\); substituting these fractions gives the Kaplan–Meier estimator. This calculation relies on censoring not selectively removing participants with different subsequent event risks, after any conditioning used in the analysis.
For arm selection, the estimand should be explicit. A fixed-horizon contrast \(S_k(\tau)-S_0(\tau)\) measures the difference in event-free probabilities at \(\tau\). It is distinct from a hazard ratio, a median survival-time difference, or a contrast in the final endpoint. A common \(\tau\) is important: comparing estimates at different horizons can make arms look different simply because the summaries refer to different periods.
Worked Example: Calculating a Fixed-Horizon PFS Contrast
Suppose a hypothetical interim analysis includes 40 participants in each of an experimental arm and control. By 6 months, 9 experimental-arm participants and 15 control participants have had a PFS event. Assume everyone has complete event status through 6 months, so there is no censoring before the horizon.
With no censoring before 6 months, the Kaplan–Meier estimates equal the observed event-free proportions. In the experimental arm, 31 of 40 participants are event-free at 6 months: $$ \widehat{S}_1(6)=\frac{31}{40}=0.775. $$ In control, 25 of 40 are event-free: $$ \widehat{S}_0(6)=\frac{25}{40}=0.625. $$
The estimated contrast is $$ \widehat{S}_1(6)-\widehat{S}_0(6)=0.775-0.625=0.150. $$ Thus, in these hypothetical data, the estimated 6-month PFS probability is 15 percentage points higher in the experimental arm. This is a descriptive interim contrast in PFS; it does not demonstrate a 15-percentage-point improvement in OS or establish final efficacy.
Worked Example: Handling Censoring in a Kaplan–Meier Estimate
Consider a hypothetical experimental arm with 20 participants. Two PFS events occur at month 2, three participants are censored at month 3, and three more PFS events occur at month 4. No other censoring or events occur before the 4-month horizon. A hypothetical control arm has 20 participants, with four PFS events at month 2 and four more at month 4, without censoring before 4 months.
In the experimental arm, 20 are at risk just before month 2, so the first factor is \(1-2/20=0.90\). The three censorings before month 4 are not events. They leave 15 participants at risk just before month 4, and the second factor is \(1-3/15=0.80\). Therefore, $$ \widehat{S}_1(4)=\left(1-\frac{2}{20}\right)\left(1-\frac{3}{15}\right) =0.90\times0.80=0.720. $$
In control, the risk sets are 20 at month 2 and 16 at month 4. Hence $$ \widehat{S}_0(4)=\left(1-\frac{4}{20}\right)\left(1-\frac{4}{16}\right) =0.80\times0.75=0.600. $$ The estimated 4-month contrast is \(0.720-0.600=0.120\), or 12 percentage points in favor of the experimental arm. The censoring assumption is essential: if those censored in the experimental arm were more likely to have an event soon afterward than those remaining under observation, the Kaplan–Meier estimate could overstate event-free probability.
Turning the Early Endpoint Into an Arm-Selection Measure
A selection rule translates early-outcome evidence into a decision about which arms continue. It should specify the endpoint, analysis horizon, estimator or test statistic, direction of benefit, and decision threshold before the interim data are examined. For a fixed-horizon event-free probability, a simple illustrative rule might retain an arm if its estimated contrast with control is at least a prespecified margin. Other rules may use uncertainty as well as the estimated contrast; the principle is the same: define the decision clearly in advance.
The rule is for selection, not a substitute for the final analysis. Selection based on an early outcome can favor arms whose interim estimates happen to be high, particularly when several arms are compared. The later final-endpoint analysis must follow the trial’s prespecified inferential plan, including any required handling of multiplicity or selection. Evidence that the early outcome predicts the final outcome on average is not enough, by itself, to justify interpreting the early treatment contrast as the final treatment effect.
Worked Example: Selecting an Arm While Keeping OS Follow-Up
Suppose a hypothetical three-arm trial has control and two experimental arms, A and B. At a prespecified 6-month PFS analysis, the estimated event-free probabilities are 0.64 for control, 0.76 for A, and 0.70 for B. The protocol’s illustrative selection rule is to retain an experimental arm if its estimated 6-month PFS advantage over control is at least 0.08.
For A, the estimated contrast is \(0.76-0.64=0.12\), or 12 percentage points. Since \(0.12\geq0.08\), A meets the rule. For B, the contrast is \(0.70-0.64=0.06\), or 6 percentage points. Since \(0.06<0.08\), B does not meet the rule. Under this specified rule, A continues and B is discontinued.
This decision does not mean A has been shown to improve OS. Participants in A and control should continue to be followed for OS under the protocol; PFS events such as progression do not end OS follow-up. The final comparison addresses the distinct OS endpoint using the prespecified final analysis. The numerical threshold here illustrates how a rule operates; a real threshold must be justified and calibrated for the design.
Assumptions, Diagnostics, and Failure Modes
Before using an early time-to-event measure, check that its definition and data support the intended comparison. The following conditions are especially important:
- Consistent endpoint ascertainment. Progression criteria, assessment schedules, event dating, and censoring rules should be applied comparably across arms. Differential assessment can create apparent differences in time to progression.
- Appropriate censoring assumptions. The Kaplan–Meier estimator assumes that, within the analysis groups or conditional strata, censoring does not depend on the unobserved event time in a way left unaccounted for. Reasons and timing of censoring should be reviewed by arm.
- Enough information by the chosen horizon. A nominal 6-month summary is unstable or uninformative if few participants have been followed near 6 months. Inspect numbers at risk and the amount of follow-up near \(\tau\); do not treat a sparse tail estimate as precise.
- A prespecified common horizon and event definition. Changing either after looking at the results can make the selection rule data-dependent and harder to interpret.
- Continued ascertainment of the final endpoint. If the final endpoint is OS, follow-up should not stop at progression simply because PFS was used for selection. Loss of OS follow-up after an interim decision can undermine the final comparison.
A useful diagnostic display includes Kaplan–Meier curves for each arm, numbers at risk at relevant times, and a summary of censoring and event counts. Look for unusually different censoring patterns, abrupt changes in assessment, or a curve that depends on only a handful of participants near the chosen horizon. These checks do not prove that assumptions hold, but they can reveal when the interim measure is not a reliable basis for a decision.
| Approach | What it contributes | Main caution for arm selection |
|---|---|---|
| Fixed-horizon event-free probability | A direct, interpretable arm contrast at a common time | Requires sufficient follow-up near the horizon and sound censoring assumptions |
| Binary event status by a cutoff | A simple comparison of event occurrence by a specified time | Can mishandle incomplete follow-up if censoring before the cutoff is treated as no event |
| Waiting for the final endpoint | Direct evidence about the endpoint that determines final success | May delay arm selection and extend trial duration |
| Early time-to-event outcome used for selection | Uses event timing while retaining a distinct final endpoint | Can select poorly if early and final effects differ; selection is not proof of surrogacy |
A binary cutoff summary can be useful when every participant has known status at that cutoff. If some participants have shorter follow-up, classifying them as event-free simply because no event has yet been recorded biases the comparison; a time-to-event method is designed to use their observed follow-up appropriately under its assumptions. Conversely, a time-to-event summary does not remove the need for sufficient follow-up or comparable event ascertainment. The best selection measure depends on the clinical question and the trial’s timing, not on a preference for a more elaborate statistic.
Common Pitfall: Treating an Early Event Outcome as the Final Endpoint
A frequent error is to reason, “PFS is better in the selected arm, so the final OS endpoint will also be better.” The conclusion does not follow from the interim contrast. Even a strong association between PFS and OS at the participant level does not guarantee that treatment effects on PFS reliably predict treatment effects on OS across arms. The correction is to describe the PFS contrast as evidence used for the prespecified selection decision, maintain follow-up for OS, and interpret final OS results using the final analysis plan.
The reverse mistake is to assume that a selection endpoint must itself be a surrogate to be useful. It need not be a replacement for OS: it can be a practical intermediate measure for deciding which arms merit continued study. Its usefulness depends on the selection objective, the quality of its measurement, and the consequences of selecting or dropping an arm. This is why the endpoint and rule should be justified before enrollment and evaluated as part of the design.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In a hypothetical trial, estimated 9-month event-free probabilities are 0.72 in an experimental arm and 0.65 in control. State the estimated contrast and interpret it without making a claim about the final endpoint.
- At a fixed horizon, why should a participant censored before that time not automatically be counted as event-free through the horizon?
- A selection rule retains an arm when its estimated event-free probability exceeds control by at least 0.05. The hypothetical estimates are 0.71 and 0.68, respectively. Does the arm meet the rule?
- If PFS guides selection but OS is the final endpoint, what should happen to OS follow-up after a participant experiences progression?
Solutions. (1) The contrast is \(0.72-0.65=0.07\), a 7-percentage-point higher estimated probability of remaining event-free through 9 months in the experimental arm. It does not establish an OS effect. (2) Censoring means the participant’s event status after the last observed time is unknown. Counting that person as event-free through the horizon would treat missing follow-up as proof of no event. (3) The contrast is \(0.71-0.68=0.03\), or 3 percentage points; because \(0.03<0.05\), the arm does not meet this rule. (4) Continue OS follow-up according to protocol; progression is a PFS event but does not itself end observation for death, the OS event.